Best Known (57−10, 57, s)-Nets in Base 7
(57−10, 57, 164710)-Net over F7 — Constructive and digital
Digital (47, 57, 164710)-net over F7, using
- net defined by OOA [i] based on linear OOA(757, 164710, F7, 10, 10) (dual of [(164710, 10), 1647043, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(757, 823550, F7, 10) (dual of [823550, 823493, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(8) [i] based on
- linear OA(757, 823543, F7, 10) (dual of [823543, 823486, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(750, 823543, F7, 9) (dual of [823543, 823493, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(70, 7, F7, 0) (dual of [7, 7, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(70, s, F7, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(9) ⊂ Ce(8) [i] based on
- OA 5-folding and stacking [i] based on linear OA(757, 823550, F7, 10) (dual of [823550, 823493, 11]-code), using
(57−10, 57, 516680)-Net over F7 — Digital
Digital (47, 57, 516680)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(757, 516680, F7, 10) (dual of [516680, 516623, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(757, 823543, F7, 10) (dual of [823543, 823486, 11]-code), using
- an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- discarding factors / shortening the dual code based on linear OA(757, 823543, F7, 10) (dual of [823543, 823486, 11]-code), using
(57−10, 57, large)-Net in Base 7 — Upper bound on s
There is no (47, 57, large)-net in base 7, because
- 8 times m-reduction [i] would yield (47, 49, large)-net in base 7, but